Calculating the Perimeter and Area of Triangles (A) - Free Printable

Here are the step-by-step solutions for each triangle problem.

1. Right Triangle (feet)

Given:
* Side $a = 8$ ft
* Side $b = 15.6$ ft
* Hypotenuse $c = 18.7$ ft

Perimeter ($P$):
Add all three sides together.
$$P = 8 + 15.6 + 18.7$$
$$P = 42.3 \text{ ft}$$
*(Note: The answer key in the image lists 44.6 ft, which appears to be incorrect based on the numbers provided. The correct sum is 42.3 ft.)*

Area ($A$):
For a right triangle, use the two legs (the sides touching the right angle) as the base and height.
$$A = \frac{1}{2} \times \text{base} \times \text{height}$$
$$A = \frac{1}{2} \times 15.6 \times 8$$
$$A = 7.8 \times 8$$
$$A = 62.4 \text{ ft}^2$$
*(Note: The answer key in the image lists 80.34 ft², which is incorrect. The correct area is 62.4 ft².)*

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2. Obtuse Triangle (cm)

Given:
* Side $a = 6$ cm
* Side $b = 7.4$ cm
* Side $c = 10.2$ cm
* Base for area calculation = $4.2$ cm
* Height for area calculation = $6$ cm (shown by the dotted line)

Perimeter ($P$):
Add the lengths of the three outer sides.
$$P = 6 + 7.4 + 10.2$$
$$P = 23.6 \text{ cm}$$
*(Note: The answer key in the image lists 21.7 cm, which is incorrect. The correct sum is 23.6 cm.)*

Area ($A$):
Use the formula $A = \frac{1}{2} \times \text{base} \times \text{height}$.
$$A = \frac{1}{2} \times 4.2 \times 6$$
$$A = 2.1 \times 6$$
$$A = 12.6 \text{ cm}^2$$
*(This matches the answer key.)*

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3. Isosceles Triangle (mi)

Given:
* Side $a = 9.6$ mi
* Side $b = 9$ mi
* Side $c = 7.4$ mi
* Height = $8.5$ mi
* Base = $7.4$ mi

Perimeter ($P$):
Add all three sides.
$$P = 9.6 + 9 + 7.4$$
$$P = 26 \text{ mi}$$
*(This matches the answer key.)*

Area ($A$):
$$A = \frac{1}{2} \times \text{base} \times \text{height}$$
$$A = \frac{1}{2} \times 7.4 \times 8.5$$
$$A = 3.7 \times 8.5$$
$$A = 31.45 \text{ mi}^2$$
*(This matches the answer key.)*

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4. Acute Triangle (cm)

Given:
* Side $a = 15.1$ cm
* Side $b = 15.2$ cm
* Side $c = 16.5$ cm
* Height = $12.8$ cm
* Base = $16.5$ cm

Perimeter ($P$):
Add all three sides.
$$P = 15.1 + 15.2 + 16.5$$
$$P = 46.8 \text{ cm}$$
*(This matches the answer key.)*

Area ($A$):
$$A = \frac{1}{2} \times \text{base} \times \text{height}$$
$$A = \frac{1}{2} \times 16.5 \times 12.8$$
$$A = 16.5 \times 6.4$$
$$A = 105.6 \text{ cm}^2$$
*(This matches the answer key.)*

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5. Right Triangle (AU)

Given:
* Side $a = 17.8$ AU
* Side $b = 14.9$ AU
* Hypotenuse $c = 28.2$ AU

Perimeter ($P$):
Add all three sides.
$$P = 17.8 + 14.9 + 28.2$$
$$P = 60.9 \text{ AU}$$
*(Note: The answer key in the image lists 48.8 AU, which is incorrect. It seems they may have forgotten to add the hypotenuse or added incorrectly. The correct sum is 60.9 AU.)*

Area ($A$):
Use the two legs as base and height.
$$A = \frac{1}{2} \times 14.9 \times 17.8$$
$$A = 14.9 \times 8.9$$
$$A = 132.61 \text{ AU}^2$$
*(Note: The answer key in the image lists 102.065 AU², which is incorrect. The correct area is 132.61 AU².)*

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6. Obtuse Triangle (yd)

Given:
* Side $a = 10.3$ yd
* Side $b = 17.1$ yd
* Side $c = 24.1$ yd
* Height = $10.3$ yd
* Base = $3.1$ yd

Perimeter ($P$):
Add the three outer sides.
$$P = 10.3 + 17.1 + 24.1$$
$$P = 51.5 \text{ yd}$$
*(Note: The answer key in the image lists 31.3 yd, which is incorrect. The correct sum is 51.5 yd.)*

Area ($A$):
$$A = \frac{1}{2} \times \text{base} \times \text{height}$$
$$A = \frac{1}{2} \times 3.1 \times 10.3$$
$$A = 1.55 \times 10.3$$
$$A = 15.965 \text{ yd}^2$$
*(Note: The answer key in the image lists 16.74 yd², which is incorrect. The correct area is 15.965 yd².)*

Final Answer:
1. P = 42.3 ft, A = 62.4 ft²
2. P = 23.6 cm, A = 12.6 cm²
3. P = 26 mi, A = 31.45 mi²
4. P = 46.8 cm, A = 105.6 cm²
5. P = 60.9 AU, A = 132.61 AU²
6. P = 51.5 yd, A = 15.965 yd²